Commentary on Proposition V.13¶
Back to Proposition V.13.
Algebraically, if
$$\begin{aligned} a : b &= c : d, \\ \text{and}\qquad c : d &> e : f, \\ \text{then}\qquad a : b &> e : f. \end{aligned}$$
After the words “for, since” in the first line of the proof, Theon added “C has to D a greater ratio than E has to F,” so that “there are some equimultiples” began, with him, the principal sentence.
The Greek text has, after “of D, F other, chance, equimultiples,” “and the multiple of C is in excess of the multiple of D….” The meaning being “such that,” I have substituted this for “and,” after Simson.
The following will show the method of Euclid’s proof.
Since
$$c : d > e : f,$$
there will be some equimultiples $mc$, $me$ of $c$, $e$, and some equimultiples $nd$, $nf$ of $d$, $f$, such that
$$mc > nd, \text{ while } me \ngtr nf.$$
But, since $a : b = c : d$, therefore, according as $ma > = < nb$, $mc > = < nd$.
And $mc > nd$ ; therefore $ma > nb$, while (from above) $me \ngtr nf$.
Simson adds as a corollary the following :
“If the first have a greater ratio to the second than the third has to the fourth, but the third the same ratio to the fourth which the fifth has to the sixth, it may be demonstrated in like manner that the first has a greater ratio to the second than the fifth has to the sixth.”
This however scarcely seems to be worth separate statement, since it only amounts to changing the order of the two parts of the hypothesis.